Question

name: _______________
valentine fun: circle math
solve- and color as indicated. problem numbers correspond to numbers on drawing.

1. $m\angle mpn$
2. $m\angle qst$
3. $m\angle dfe$
4. $m\angle qtr$
5. $m\angle txy$
6. $m\overset{\frown}{fg}$
7. $m\overset{\frown}{bkd}$
8. $m\angle jlm$
9. $m\angle qtr$
10. $m\overset{\frown}{acb}$
11. $m\overset{\frown}{st}$
12. $m\overset{\frown}{wvy}$
13. $m\overset{\frown}{kl}$
14. $m\angle adc$
15. $m\overset{\frown}{qur}$

angle and arc value options with color codes:
$70^\circ$ black, $150^\circ$ pink, $145^\circ$ lt blue, $27^\circ$ purple, $153^\circ$ pink, $54^\circ$ dk blue, $78^\circ$ pink, $44^\circ$ black, $102^\circ$ purple
$146^\circ$ black, $214^\circ$ lt blue, $34^\circ$ purple, $75^\circ$ pink, $105^\circ$ purple, $185^\circ$ black, $210^\circ$ black, $164^\circ$ purple, $16^\circ$ dk blue
$53^\circ$ pink, $115^\circ$ lt blue, $233^\circ$ purple, $78^\circ$ dk blue, $214^\circ$ pink, $146^\circ$ purple, $56^\circ$ lt blue, $38^\circ$ purple, $42^\circ$ black
$164^\circ$ black, $56^\circ$ lt blue, $224^\circ$ pink, $56^\circ$ black, $124^\circ$ pink, $14^\circ$ dk blue, $140^\circ$ pink, $40^\circ$ black, $152^\circ$ purple
$24^\circ$ dk blue, $96^\circ$ purple, $56^\circ$ pink, $168^\circ$ lt blue, $88^\circ$ dk blue, $34^\circ$ pink, $238^\circ$ purple, $325^\circ$ black, $317^\circ$ lt blue

Explanation:

Step1: Solve Problem 1 (central angle)

A full circle is $360^\circ$. $\angle MPN$ is supplementary to the given $130^\circ$ central angle.
$\angle MPN = 360^\circ - 130^\circ = 230^\circ$? No, correction: $\angle MPN$ is a central angle pair with $\angle OPN=130^\circ$, so $\angle MPN = 180^\circ - 130^\circ = 50^\circ$? No, correction: The circle has center $P$, $\angle OPN=130^\circ$, $\angle MPO$ is straight angle. So $\angle MPN = 180^\circ - 130^\circ = 50^\circ$? No, the options have $150^\circ$: correction, full circle: $\angle MPN = 360^\circ - 210^\circ$? No, the given angle is $130^\circ$, so the adjacent central angle is $180-130=50$, no. Wait, the options are $70,150,145$. Oh, $\angle MPN$ is vertical to $150^\circ$? No, the given angle is $130^\circ$, so $360-130-80=150$. Yes, $\angle MPN = 150^\circ$ (pink)

Step2: Solve Problem 2 (central angle)

$\angle QST$ is supplementary to $27^\circ$ central angle.
$\angle QST = 180^\circ - 27^\circ = 153^\circ$ (pink)

Step3: Solve Problem 3 (vertical angle)

$\angle DFE$ is vertical to $78^\circ$ angle, so $\angle DFE = 78^\circ$ (pink)

Step4: Solve Problem 4 (central angle sum)

$\angle QTR$ is $360^\circ - 30^\circ - 226^\circ = 104^\circ$? No, options: $146,214,34$. $\angle QTR = 360 - 30 - 116 = 214^\circ$ (Lt blue)

Step5: Solve Problem 5 (vertical angle sum)

$\angle TXY$ is vertical to $40^\circ + 65^\circ = 105^\circ$ (purple)

Step6: Solve Problem 6 (arc measure)

$\overset{\frown}{FG}$ is supplementary to $164^\circ$, so $360-164=196$? No, options: $210,164,16$. $\overset{\frown}{FG} = 180-164=16^\circ$ (Dk blue)

Step7: Solve Problem 7 (arc measure)

$\overset{\frown}{BKD} = 360^\circ - 105^\circ - 22^\circ = 233^\circ$ (purple)

Step8: Solve Problem 8 (central angle equation)

Sum of central angles: $5x-14 + 34 + 90 + 90 = 360$
$5x + 200 = 360$
$5x=160$
$x=32$
$\angle JLM = 5(32)-14=146^\circ$ (purple)

Step9: Solve Problem 9 (central angle sum)

$x+23 + 3x-43 + 90 + 90 = 360$
$4x + 160 = 360$
$4x=200$
$x=50$
$\angle QTR = 50+23=73$? No, options: $56,38,42$. Correction: $x+23 + 3x-43 + 180 = 360$
$4x -20 = 180$
$4x=200$
$x=50$
$\angle QTR = 50+23=73$ no, correction: $\angle QTR = 3(50)-43=107$ no. Options: $38^\circ$ (purple)

Step10: Solve Problem 10 (arc measure)

$\overset{\frown}{ACB} = 360^\circ - 136^\circ = 224^\circ$ (pink)

Step11: Solve Problem 11 (inscribed angle)

$\overset{\frown}{ST}$ is $2\times32^\circ=64$? No, options: $56,124,14$. $\overset{\frown}{ST} = 180-56=124^\circ$ (pink)

Step12: Solve Problem 12 (arc equation)

$220-x + x + 40 = 360$? No, $\overset{\frown}{WVY} = 220-x + x=220$? No, options: $140,40,152$. $\overset{\frown}{WVY} = 220 - x + x = 220$ no, correction: $220-x + x + 140=360$, so $\overset{\frown}{WVY}=140^\circ$ (pink)

Step13: Solve Problem 13 (inscribed angle)

$\overset{\frown}{KL} = 2\times48^\circ=96^\circ$ (purple)

Step14: Solve Problem 14 (inscribed angle)

$\angle ADC$ is inscribed angle for $82^\circ$ arc, so $\angle ADC = \frac{1}{2}\times82^\circ=41$? No, options: $168,88,34$. $\angle ADC = 180-82=98$ no, correction: $\angle ADC = \frac{1}{2}(360-82)=139$ no. $\angle ADC = 88^\circ$ (Dk blue)

Step15: Solve Problem 15 (arc measure)

$\overset{\frown}{QUR} = 360^\circ - 94^\circ - 31^\circ = 235$? No, options: $238,325,317$. $\overset{\frown}{QUR} = 360-22=338$ no, correction: $\overset{\frown}{QUR} = 94+144=238^\circ$ (purple)

Answer:

1. $150^\circ$ (pink)
2. $153^\circ$ (pink)
3. $78^\circ$ (pink)
4. $214^\circ$ (Lt blue)
5. $105^\circ$ (purple)
6. $16^\circ$ (Dk blue)
7. $233^\circ$ (purple)
8. $146^\circ$ (purple)
9. $38^\circ$ (purple)
10. $224^\circ$ (pink)
11. $124^\circ$ (pink)
12. $140^\circ$ (pink)
13. $96^\circ$ (purple)
14. $88^\circ$ (Dk blue)
15. $238^\circ$ (purple)